O Sleepless as the river under thee, Vaulting the sea, the prairies' dreaming sod,
The speaker invokes the river one more time: "O Sleepless," "thee."
Like the East River that it spans, the Brooklyn Bridge never seems to rest. There is always traffic crossing it.
The East River connects to the Atlantic Ocean, so technically the Brooklyn Bridge goes across ("vaults") the sea.
On a symbolic level, the bridge connects all of America, from East Coast to West Coast, through the Great Plains of middle America.
The soil ("sod") of the plains is described as "dreaming." At least someone is getting some sleep around here!
Middle America is a place of dreams, and the bridge is a symbol of unity.
Hart Crane loved puns, so we have to consider the erotic interpretation of these lines. An archaic word (and insult) for gay men is "sodomites," and Crane came from the land of the prairies, rural Ohio. So in that sense he is "the prairies' dreaming sod" that the bridge spans (source).
Unto us lowliest sometime sweep, descend And of the curveship lend a myth to God.
The speaker asks one final favor of the great Brooklyn Bridge. Pretty please?
He wants the bridge to "descend" to the level of the common man, the "lowliest" Americans.
No, he doesn't want the bridge to collapse! He just wants ordinary people to be able to see the bridge imaginatively for the symbol of unity and mysticism that it is.
He wants the bridge to "lend a myth to God" – to give God a hand in inspiring people.
The last line is another paradox. God is already mythic, not necessarily in the sense of "not true," but in the sense of explaining the way things are in a very grand and imaginative way.
The bridge is a secular, non-religious symbol. It was created by man for practical use. But it can help fill the gap that religion has left behind in the modern world.
"Curveship" is an invented word. It could refer to the bridge itself, similar to the word "lordship," as when characters in old English dramas say "Your Lordship" to royalty.
The bridge's defining characteristic is the sweeping "curves" of its suspenders.
For Crane, the bridge provides a powerful, secular mythology to replace the outdated mythology of pre-modern religions.